Question: $ -2.\overline{2} \div 0.\overline{3} = {?} $
Solution: First convert the repeating decimals to fractions. $\begin{align*} 10x &= -22.2223...\\ x &= -2.2223...\end{align*} $ $\begin{align*} 9x &= -20 \\ x &= -\dfrac{20}{9}\end{align*} $ $\begin{align*} 10y &= 3.3333...\\ y &= 0.3333...\end{align*} $ $\begin{align*} 9y &= 3 \\ y &= \dfrac{3}{9}\end{align*} $ So, the problem becomes: $ -\dfrac{20}{9} \div \dfrac{3}{9} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{20}{9} \times \dfrac{9}{3} = {?} $ $ \phantom{-\dfrac{20}{9} \times \dfrac{3}{9}} = \dfrac{-20 \times 9}{9 \times 3} $ $ \phantom{-\dfrac{20}{9} \times \dfrac{3}{9}} = \dfrac{-20 \times \cancel{9}} {\cancel{9} \times 3} $ $ \phantom{-\dfrac{20}{9} \times \dfrac{3}{9}} = -\dfrac{20}{3} $